Given a compact set of real numbers, a random $C^{m + \alpha}$ -diffeomorphism is constructed such that the image of any measure concentrated on the set and satisfying a certain condition involving a real number $s$ , almost surely has Fourier dimension greater than or equal to $s / (m + \alpha)$ . This is used to show that every Borel subset of the real numbers of Hausdorff dimension $s$ is $C^{m + \alpha}$ -equivalent to a set of Fourier dimension greater than or equal to $s / (m + \alpha )$ . In particular every Borel set is diffeomorphic to a Salem set, and the Fourier dimension is not invariant under $C^{m}$ -diffeomorphisms for any $m$ .

Over time, a population acquires neutral genetic substitutions as a consequence of random drift. A famous result in population genetics asserts that the rate, K, at which these substitutions accumulate in the population coincides with the mutation rate, u, at which they arise in individuals: K = u. This identity enables genetic sequence data to be used as a “molecular clock” to estimate the timing of evolutionary events. While the molecular clock is known to be perturbed by selection, it is thought that K = u holds very generally for neutral evolution. Here we show that asymmetric spatial population structure can alter the molecular clock rate for neutral mutations, leading to either K<u or K>u. Our results apply to a general class of haploid, asexually reproducing, spatially structured populations. Deviations from K = u occur because mutations arise unequally at different sites and have different probabilities of fixation depending on where they arise. If birth rates are uniform across sites, then K
u. In general, K can take any value between 0 and Nu. Our model can be applied to a variety of population structures. In one example, we investigate the accumulation of genetic mutations in the small intestine. In another application, we analyze over 900 Twitter networks to study the effect of network topology on the fixation of neutral innovations in social evolution.

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For a sub-Riemannian manifold provided with a smooth volume, we relate the small-time asymptotics of the heat kernel at a point y of the cut locus from x with roughly “how much” y is conjugate to x. This is done under the hypothesis that all minimizers connecting x to y are strongly normal, i.e. all pieces of the trajectory are not abnormal. Our result is a refinement of the one of Leandre 4t log pt(x, y) ! −d2(x, y) for t ! 0, in which only the leading exponential term is detected. Our results are obtained by extending an idea of Molchanov from the Riemannian to the sub-Riemannian case, and some details we get appear to be new even in the Riemannian context. These results permit us to obtain properties of the sub-Riemannian distance starting from those of the heat kernel and vice versa. For the Grushin plane endowed with the Euclidean volume, we get the expansion pt(x, y)  t−5/4 exp(−d2(x, y)/4t) where y is reached from a - Riemannian point x by a minimizing geodesic which is conjugate at y.

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